HNN decomposition of finite rank free group over infinite rank subgroups

Question

It's a nice result of Swarup that whenever a free group $G$ splits as an HNN extension $G = J \ast_{H,t}$ with $H$ a finitely generated subgroup, there exist splittings $J = J_1 \ast J_2$ and $H = H_1 \ast H_2$ such that $H_1 \neq 1$ is a free factor of $J_1$.

I would like to know if there exists an analogous result in the literature in the case when $G$ is a finite rank free group which admits an HNN splitting $G = J \ast_H$, where $J$ and $H$ are infinite rank free groups? Or conversely if there's an obvious counter-example to such a result.

Answer 1

You haven't quite given the full strength of Swarup's result, which is what makes it so useful: moreover, $H_2$ and $tHt^{-1}$ are conjugate into $J_2$. The geometric interpretation is that a natural square-complex realisation of $G$ has a free face.

There is no analogous result of the full statement of Swarup's theorem when the edge and vertex groups are of infinite rank. For example, take $G=\langle a,t\rangle$ and consider the retraction to $\mathbb{Z}=\langle t\rangle$ given by killing $a$. This gives us the semidirect product decomposition

$G=J\rtimes \mathbb{Z}$

where $J$ is free on $\{t^nat^{-n}\mid n\in\mathbb{Z}\}$. We can think of this as an HNN decomposition with vertex and both edge groups equal to $J$. Thus, we can't have one of the edge groups conjugate into a proper free factor of $J$, so the analogue of Swarup's theorem doesn't hold.

However, off the top of my head, I don't know if your weaker version of Swarup might work in the infinite rank case. (Note: In an earlier version I said it didn't, but I had misinterpreted what you wrote slightly. Apologies!)

By the way, there are many modern generalisations of Swarup's theorem. You might be interested in this paper of Diao and Feighn, which explains the square-complex point of view.

Answer 2

The situation with infinite rank edge and vertex groups is qualitatively very different. For example, the following paper by Hagen and Wise shows that a non-trivial class of hyperbolic groups (constructible groups) are virtually HNN extensions of (infinite rank) free groups Special groups with an elementary hierarchy are virtually free-by-ℤ.

On a technical note, in Swarup's original paper, the paper by Diao and Feighn mentionned in another answer, and my own paper On the one-endedness of graphs of groups which is inspired by Diao-Feighn, finite generation of the edge groups plays a critical role. In the more modern papers, finite generation of edge groups ensures that the group acts co-compactly on a nice square complex.

All this to say that a generalization of Swarup's Theorem to your situation may require to be some new ideas.